Topology Proceedings CLASSIFICATION OF FINITE ALEXANDER QUANDLES

Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t]-submodules Im(1− t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Zn[t ]/(t− a) where gcd(n, a) = 1 (called linear quandles) are isomorphic, as well as specific conditions on when two linear quandles are dual and which linear quandles are connected. We apply this result, obtaining a procedure for classifying Alexander quandles of any finite order and as an application we list the numbers of distinct and connected Alexander quandles with up to fifteen elements.